The motivation for introducing the twisted eigenvalue problem is the lack of <<prehomogeneity>> of the space \(\displaystyle \big (\mathbb {P}(\mathrm {J}_{3}(\mathbb {O})), \mathrm {SO}_7 \times \mathrm {SL}_3 \times \mathbb {C}^*, \mathrm {S_{ODM}}\big )\) . This lack of prehomogeneity is partly explained by the relatively big codimension of the subgroup of \(\displaystyle \mathrm {E}_{6}\)  generated by \(\displaystyle \mathrm {SO}_{7}\)  and \(\displaystyle \mathrm {SL}_3\) . On the other hand, it is well-known (see chapter \(\displaystyle 14\)  of [1]}) that \(\displaystyle \mathrm {E}_6\)  is generated by \(\displaystyle \mathrm {Spin}_8\)  and \(\displaystyle \mathrm {SL}_{3}\) . Thus, one could hope for an eigenvalue problem having a larger group of symmetries. As it happens, the symmetry group of the twisted octonionic eigenvalue problem contains the subgroup of \(\displaystyle \mathrm {SL}_{27}\)  generated by \(\displaystyle \mathrm {Spin}_{7}\)  and \(\displaystyle \mathrm {SL}_3\)  The action of \(\displaystyle \mathrm {Spin}_7\)  on \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\)  that we exhibit is not the standard one and does not factor through an embedding of \(\displaystyle \mathrm {Spin}_7\)  in \(\displaystyle \mathrm {E}_6\) . As a matter of fact, we haven't been able to identify the subgroup of \(\displaystyle \mathrm {SL}_{27}\)  generated by \(\displaystyle \mathrm {Spin}_{7}\)  and \(\displaystyle \mathrm {SL}_3\) . and we prove that \(\displaystyle \mathrm {J}_{3}(\mathbb {O})\)  is prehomogeneous for this action. We also show that this twisted problem is singular on the union of the Cartan cubic and a sextic hypersurface. To the best of our knowledge, this geometry has never appeared before in the literature.

